Find The Measure of A Positive Angle Coterminal Calculator
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Coterminal angles are angles that share the same terminal side. They differ by integer multiples of 360°. This calculator helps you find the smallest positive angle coterminal to any given angle.
What are Coterminal Angles?
Coterminal angles are angles that have the same terminal side when drawn in standard position. This means they differ by full rotations (360°) of each other. For example, 45° and 405° are coterminal because 405° - 360° = 45°.
Coterminal angles are useful in trigonometry, navigation, and engineering where angles are measured from different starting points but represent the same direction.
How to Find a Positive Coterminal Angle
To find the smallest positive angle coterminal to a given angle:
- If the angle is already positive and less than 360°, it is its own coterminal angle.
- If the angle is negative, add 360° repeatedly until you get a positive angle.
- If the angle is greater than 360°, subtract 360° repeatedly until you get an angle between 0° and 360°.
This process ensures you find the smallest positive angle that is coterminal with the original angle.
Formula
If θ_coterminal is negative, add 360° to get the positive coterminal angle.
Where:
- θ is the original angle in degrees
- θ_coterminal is the smallest positive coterminal angle
- mod is the modulo operation (remainder after division)
Example Calculation
Let's find the smallest positive angle coterminal to -100°:
- Calculate -100° mod 360° = -100°
- Since the result is negative, add 360°: -100° + 360° = 260°
- The smallest positive coterminal angle is 260°
You can verify this by checking that 260° - (-100°) = 360°, which is a full rotation.
FAQ
What is the difference between coterminal and supplementary angles?
Coterminal angles share the same terminal side and differ by full rotations (360°). Supplementary angles add up to 180° and are on a straight line. They are not the same concept.
Can coterminal angles be negative?
Yes, coterminal angles can be negative. However, the smallest positive coterminal angle is typically what's needed in most applications.
How do coterminal angles relate to trigonometric functions?
Coterminal angles have identical trigonometric function values because they terminate at the same point on the unit circle. This property is fundamental in trigonometry.